L-BFGS Research Articles

CSEM Optimization Using the Correspondence Principle

Traditionally, 3D modeling of marine controlled-source electromagnetic (CSEM) data (in the frequency domain) involves high-memory demand, requiring solving a large linear system for each frequency. To address this problem, we propose to solve Maxwell’s equations in a fictitious dielectric medium with time-domain finite-difference methods, with the support of the correspondence principle. As an advantage of this approach, we highlight the possibility of its implementation for execution with GPU accelerators, in addition to multi-frequency data modeling with a single simulation. Furthermore, we explore using the correspondence principle to the inversion of CSEM data by calculating the gradient of the least-squares objective function employing the adjoint-state method to establish the relationship between adjoint fields in a conductive medium and their counterparts in the fictitious dielectric medium, similar to the approach used in forward modeling. We validate this method through 2D inversions of three synthetic CSEM datasets, computed for a simple model consisting of two resistors in a conductive medium, a model adapted from a CSEM modeling and inversion package, and the last one based on a reference model of turbidite reservoirs on the Brazilian continental margin. We also evaluate the differences between the results of inversions using the steepest descent method and our proposed momentum method, comparing them with the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm (L-BFGS-B). In all experiments, we use smoothing by model reparameterization as a strategy for regularizing and stabilizing the iterations throughout the inversions. The results indicate that, although it requires more iterations, our modified momentum method produces the best models, which are consistent with results from the L-BFGS-B algorithm and require less storage per iteration.

Optimized neural network-based model to predict the shear strength of trapezoidal-corrugated steel webs

Beam-like members use corrugated webs to increase their shear strength, stability, and efficiency. The corrugation positively affects the members' structural characteristics, especially those governed by the web parameters, such as the shear strength, while reducing the total weight. Existing code and analytical models for predicting the shear strength of trapezoidal corrugated steel webs (TCSWs) are summarized. This paper presents an optimized Artificial Neural Network (ANN)-based model to estimate the shear strength of steel girders with a TCSW subjected to a concentrated force. A database of 206 experimental results from the literature is used to feed the ANNs. Six geometrical and material parameters were identified as input variables, and the experimental shear strength at failure was considered the output variable. Four hyperparameter optimization techniques are applied to refine the ANN models: Bayesian Optimization (BO), Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), Firefly Algorithm (FA), and African Buffalo Optimization (ABO). The performance metrics indicate that the ABO-ANN model is the most effective among these. The predictions of the developed ML model were also compared with those of existing code and analytical models. The comparisons illustrated that the ANN-based model outperforms the other existing models. The sensitivity analysis using the proposed ANN-based model captured the relationships and interactions among the geometric and material parameters and their impact on shear strength. One main finding is that the corrugation angle in the 35–45° range maximized the TCSW shear strength.

Enhanced fast inertial relaxation engine (FIRE) for multiscale simulations

In multiscale modeling methods (MMM), the integration of atomistic to continuum coupling is a common practice, where two regions have their own distinct length scales. The equilibrium configurations of such multiscale systems under given conditions are typically obtained through energy minimization algorithms (EMA). However, traditional EMAs, such as the conjugate gradient (CG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithms, are unable to discern the diverse scales inherent in such systems. In this work, it is found that the convergence rate of energy minimization in multiscale simulations is significantly slower than that in full atomistic simulations, regardless of using CG, LBFGS algorithms or the latest fast inertial relaxation engine (FIRE). The lower efficiency emerges due to the coexistence of atoms and nodes with distinct length scales within the multiscale framework, yet the current EMAs fail to differentiate between them. It results in disparate convergence rates across different scales, which undermines both computational accuracy and efficiency. To address the issue, a multiscale FIRE algorithm which updates positions of atoms and nodes synchronously by employing appropriate effective mass is proposed. The optimal effective mass is determined by synchronizing the vibration of harmonic oscillators across different scales. By employing the multiscale FIRE algorithm, the computational efficiency increased by 24.4 and 23.7 times compared to the CG and LBFGS algorithms when used for multiscale nanoindentation simulations. These findings and the proposed algorithm provide valuable insights for structural relaxations of multiscale physical problems and are promising to further improve the computational accuracy and efficiency of MMMs.

Wave impedance inversion and interpretation of Ground Penetrating Radar (GPR) data for Amery Ice Shelf in East Antarctica

The polar ice sheet is vital for the global climate system, influencing the Antarctic ice sheet's mass balance, sea level rise, and Earth's surface energy. Accurate measurement of its thickness and internal structure is imperative for comprehending glacier evolution and climate change. Ground Penetrating Radar (GPR) is a high-resolution and non-invasive exploration instrument that is extensively used for glacier detection, but only limited geometric information can be identified in the GPR profile, making it difficult to capture the quantitative distribution of physical parameters. To image the fine dielectric parameters of GPR data acquired from the Amery Ice Shelf in East Antarctica in 2003, we propose a multi-trace GPR impedance inversion approach. This method utilizes the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm for GPR data, coupled with weighted L2-norm total-variation multiplicative regularization (MR) to quantitatively estimate the dielectric parameters in the interior of the ice sheet. This scheme adaptively adapts regularization parameters, enhances vertical resolution, and suppresses noise. Additionally, considering lateral correlation, we integrate directional differences to enhance lateral continuity. After validating with complex test data, we apply the method to Amery Ice Shelf GPR data, quantitatively imaging the ice sheet's geometric and dielectric parameters. The inversion results reveal ice thickness, internal features, and the interface between freshwater ice and refrozen sea ice. This allowed us to determine the ablation and refreezing zone boundary at the bottom of ice shelf, providing insights into melting/freezing mechanisms, ice shelf stability, and simulating ice shelf interior dynamics.

Prediction of adsorption performance of ZIF-67 for malachite green based on artificial neural network using L-BFGS algorithm

Given the necessity and urgency in removing organic pollutants such as malachite green (MG) from the environment, it is vital to screen high-capacity adsorbents using artificial neural network (ANN) methods quickly and accurately. In this study, a series of ZIF-67 were synthesized, which adsorption properties for organic pollutants, especially MG, were systematically evaluated and determined as 241.720 mg g−1 (25 ℃, 2 h). The adsorption process was more consistent with pseudo-second-order kinetics and Langmuir adsorption isotherm, which correlation coefficients were 0.995 and 0.997, respectively. The chemisorption mechanism was considered to be π-π stacking interaction between imidazole and aromatic ring. Then, a Python-based neural network model using the Limited-memory BFGS algorithm was constructed by collecting the crucial structural parameters of ZIF-67 and the experimental data of batch adsorption. The model, optimized extensively, outperformed similar Matlab-based ANN with a coefficient of determination of 0.9882 and mean square error of 0.0009 in predicting ZIF-67 adsorption of MG. Furthermore, the model demonstrated a good generalization ability in the predictive training of other organic pollutants. In brief, ANN was successfully separated from the Matlab platform, providing a robust framework for high-precision prediction of organic pollutants and guiding the synthesis of adsorbents.

Performance analysis of un-doped and doped titania (TiO ) as an electron transport layer (ETL) for perovskite solar cells.

Density functional theory (DFT) calculations are carried out on pure and doped rutile TiO . The bandgap (E ) for pristine, S-doped, Fe-doped, and Fe/S co-doped materials is direct, with values of 2.98 eV, 2.18 eV, 1.58 eV, and 1.40 eV. The effective mass of charge carriers (m*) and ratio of effective masses of holes to effective masses of electrons (R) are also investigated, and it is discovered that Fe/S co-doped materials have the lowest charge carrier recombination rate. The Fe/S co-doped material has the highest . of doped materials shifted into the visible range. Due to the high dopant concentration in Fe and Fe/S-doped cases, the E is lowered to a relatively small value; hence, only pristine and S-doped materials are verified as electron transport layer (ETL). A solar cell device analysis employing pure and S-doped rutile TiO as ETL is completed using DFT-derived parameters in SCAPS-1D modeling software for the first time. For the optimized solar cells, current-voltage (IV) characteristics, quantum efficiency (QE), capacitance-voltage (CV) characteristics, and capacitance-frequency (Cf) characteristics are provided. The aim of the present study is to improve efficiency of perovskite solar cell by doping as well as to improve accuracy of simulation by applying DFT extracted parameters as input. From the analysis, improvement is found in efficiency of doped TiO compared to un-doped TiO . The efficiency of the PSC with S-doped ETL is 1.418% higher than the PSC with un-doped ETL. Quantumwise Automistic Tool Kit (ATK) is used to extract DFT parameters. Using these DFT parameters as input in SCAPS-1D (Solar Cell Capacitance Simulator), solar cells for doped and un-doped material are simulated. The density functional theory (DFT)-based orthogonalized linear combination of atomic orbital (OLCAO) technique is used. Structural optimization is done using the LBFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno). PBESol-GGA (Perdew-Burke-Ernzerhof solid-generalized gradient approximation) is applied as exchange correlation for calculating structural parameters, while MGGA-TB09 (meta-generalized gradient approximation-Tran and Blaha) is applied as exchange correlation for calculating optical and electronic properties.

Optimizing Parameter Efficiency in Machine Learning Models: A Focus on Reducing Memory Overhead with L-BFGS-Optimized Algorithms

This research addresses the critical challenge of detecting cotton leaf diseases through an advanced image processing and machine learning approach. The study primarily focuses on the extraction of significant features from cotton leaf images to facilitate accurate disease identification. The novelty of this work lies in the application of the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm for optimizing Support Vector Machines (SVM), enhancing the precision of disease detection in cotton leaves. This research method involves a comprehensive process of acquiring high-resolution leaf images, followed by an effective feature extraction technique that isolates crucial characteristics indicative of various plant diseases. Subsequently, these features are employed to train an SVM model optimized by the L-BFGS algorithm, a decision process that marks a significant departure from traditional gradient descent methods in SVM training. A pivotal aspect of this study is the comparative analysis conducted between the proposed L-BFGS-optimized SVM model and prevalent machine learning models including Random Forest, Decision Tree, Regression Models, and K-Nearest Neighbors (KNN). This comparison is grounded on various performance metrics such as accuracy, precision, recall, and F1-score, offering a comprehensive evaluation of each model's effectiveness in disease detection. The results of this research are expected to demonstrate the superiority of the L-BFGS-optimized SVM in terms of accuracy and efficiency in detecting cotton leaf diseases. This advancement holds substantial promise for agricultural technology, potentially leading to more informed and timely decision-making in crop management and disease prevention. The findings of this study aim to contribute significantly to the field of agricultural informatics, particularly in the domain of plant disease detection and management, by providing a more robust, accurate, and efficient tool for farmers and agriculturalists.

Orthogonal extended infomax algorithm.

Objective.The extended infomax algorithm for independent component analysis (ICA) can separate sub- and super-Gaussian signals but converges slowly as it uses stochastic gradient optimization. In this paper, an improved extended infomax algorithm is presented that converges much faster.Approach.Accelerated convergence is achieved by replacing the natural gradient learning rule of extended infomax by a fully-multiplicative orthogonal-group based update scheme of the ICA unmixing matrix, leading to an orthogonal extended infomax algorithm (OgExtInf). The computational performance of OgExtInf was compared with original extended infomax and with two fast ICA algorithms: the popular FastICA and Picard, a preconditioned limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm belonging to the family of quasi-Newton methods.Main results.OgExtInf converges much faster than original extended infomax. For small-size electroencephalogram (EEG) data segments, as used for example in online EEG processing, OgExtInf is also faster than FastICA and Picard.Significance.OgExtInf may be useful for fast and reliable ICA, e.g. in online systems for epileptic spike and seizure detection or brain-computer interfaces.

Three-dimensional finite-memory quasi-Newton inversion of the magnetotelluric based on unstructured grids

Abstract Simulation optimization of complex geological bodies is a necessary means to improve inversion accuracy and computational efficiency; thus, inversion of magnetotelluric (MT) based on unstructured grids has become a research hotspot in recent years. This article realizes the three-dimensional (3D) finite element forward modeling of MT based on the magnetic vector potential-electric scalar potential method, using unstructured grids as the forward modeling grid, which improves computational efficiency. The inversion uses the limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) method, and in the process of calculating the objective function gradient, the quasi-forward method is used to avoid solving the Jacobian matrix, which has the advantages of requiring small storage space and fast computational efficiency. Finally, the 3D LBFGS inversion algorithm of MT based on unstructured grids was realized, and the inversion studies of classic and complex models verified the effectiveness and the reliability of the algorithm proposed in this article.

MTAINV3D: A Three-dimensional Adaptive Magnetotelluric Inversion Tool Using Unstructured Grids

Currently, 3D magnetotelluric inversion often uses a fixed inversion grid. Too sparse inversion grids cannot approximate the complex subsurface conductivity distribution, and may also lead to inversion results failing to converge. Too dense inversion grids will increase not only the computational cost, but also the non-uniqueness of the inversion. Aiming at the above problems, we have developed a three-dimensional magnetotelluric adaptive inversion strategy with unstructured finite element and limited memory BFGS (L-BFGS) algorithms. Firstly, based on the unstructured finite element method, the high accuracy forward responses of the 3D geo-electric model with arbitrary complex terrain and conductivity distributions can be obtained. Then, the forward and inversion meshes are decoupled by using the nested forward and inversion tetrahedral meshes. The Thikhonov regularization objective function with smooth constraints is established and minimized by L-BFGS optimization algorithm with inexact line search procedure, where the gradient of the objective function is solved by using the adjoint principle. Most importantly, an inversion grid adjustment strategy based on the dual constraints of sensitivity matrix and model gradient is proposed, which are used to guide the automatic refinement of inversion grid in the optimization process. Finally, a topographic model is used to validate and test the performance of the proposed adaptive inversion algorithm.

Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations

In the past decades, multiple shooting methods have proven to be a promising direction to speed up the optimization process, especially in the context of ODE-based optimization. Very recently, Fang et al. (Journal of Computational Physics, vol. 452, 110926, 2022) proposed a multiple shooting algorithm for large-scale PDE-constrained optimization. The current paper continues this line of work and explores the potential of multiple shooting methods for optimal control problems governed by the three-dimensional Navier–Stokes equations. Similar to Fang et al., the augmented Lagrangian (AL) method is used to solve the resulting equality-constrained optimization problem, and we employ the classical limited-memory BFGS method for the unconstrained subproblems inside the AL loop. In the current work, we exploit the multiple shooting paradigm in full by processing the shooting windows parallel-in-time, allowing for significant parallel speed-ups compared to single shooting. The proposed method is validated on a velocity tracking case, using up to 100 windows. Our analysis shows that the multiple shooting algorithm allows for considerable algorithmic and parallel speed-ups. While algorithmic speed-up depends on the exact tracking case and initialization of the shooting windows, the multiple shooting algorithm always outperforms single shooting in terms of computational time (if the number of windows is sufficiently high) due to the parallel-in-time implementation. For a given amount of resources, we also show that the proposed parallel-in-time strategy can outperform spatial parallelization alone, especially when the spatial parallelization is saturated.

An Efficient Mapping Scheme on Neural Networks for Linear Massive MIMO Detection

For massive multiple-input multiple-output (MIMO) communication systems, simple linear detectors such as zero forcing (ZF) and minimum mean square error (MMSE) can achieve near-optimal detection performance with reduced computational complexity. However, such linear detectors always involve complicated matrix inversion, which will suffer from high computational overhead in the practical implementation. Due to the massive parallel-processing and efficient hardware-implementation nature, the neural network has become a promising approach to signal processing for the future wireless communications. In this paper, we first propose an efficient neural network to calculate the pseudo-inverses for any type of matrices based on the improved Newton's method, termed as the PINN. Through detailed analysis and derivation, the linear massive MIMO detectors are mapped on PINNs, which can take full advantage of the research achievements of neural networks in both algorithms and hardwares. Furthermore, an improved limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton method is studied as the learning algorithm of PINNs to achieve a better performance/complexity trade-off. Simulation results finally validate the efficiency of the proposed scheme.

Three-Dimensional Limited-Memory BFGS Inversion of Magnetic Data Based on a Multiplicative Objective Function

At present, the traditional magnetic three-dimensional inversion method has been fully developed and is widely used. Magnetic exploration is a kind of geophysical exploration method that uses the magnetic field changes (magnetic anomalies) caused by the magnetic differences between various rocks in the crust to find useful mineral resources and study the underground structure. Traditional magnetic three-dimensional inversion is relatively inefficient. Moreover, the traditional additive objective function (data fitting difference term plus regularization term and logarithmic obstacle term), which causes the regularization factor selection to be more complicated, is implemented in this method. Therefore, it is necessary to establish a new efficient three-dimensional magnetic inversion algorithm and optimize the selection of regularization factors. In this paper, based on the limited-memory BFGS (L-BFGS) method, the three-dimensional magnetic inversion of a multiplicative objective function is realized. The inversion test is conducted in this paper using both theoretical synthesis data and measured data. The results demonstrate that the limited-memory BFGS method significantly enhances the inversion efficiency and yields superior inversion outcomes compared to traditional magnetic three-dimensional inversion methods. Additionally, the multiplicative objective function-based three-dimensional magnetic inversion method simplifies the process of selecting weight factors for regularization terms.

An improved inversion method for shale elastic parameters measurement based on ultrasonic resonance spectroscopy

The accurate measurement of elastic parameters of shale cores has important guidance for the exploration and development of shale oil and gas. The traditional shale acoustic experiment requires multiple samples from different angles and its preparation process is complex. Resonant Ultrasound Spectroscopy has the advantages of repeatable measurement, high accuracy, and the ability to obtain all elastic parameters from a single small sample. However, the influence of clamping force and measuring angle needs comprehensive analysis to ensure that formant can be measured as many and accurately as possible. This research first conducts sensitivity analysis on the elastic parameters of different shale to guide the experiment. In addition, due to the more complex inversion objective function of the anisotropic shale compared to isotropic samples, the traditional Levenberg–Marquardt method has a strong dependence on the initial value and is prone to falling into local optimal solution. The L-BFGS (limited memory BFGS) method is introduced into the inversion of anisotropic elastic parameters of shale, which can significantly improve the inversion efficiency. Finally, the robustness and reliability of the inversion method are validated through theoretical and experimental measurement results. [Work supported by the National Natural Science Foundation of China (42004117 and 42127807).]

A Novel Planning and Delivery Technology: Dose, Dose Rate and Linear Energy Transfer (LET) Optimization Based on Spot-Scanning Proton Arc Therapy FLASH (SPLASHLET)

To achieve a high conformal dose with Linear Energy Transfer (LET) optimized FLASH proton therapy, we introduced a new planning and delivery technique concept, the voxel-wised optimization of LET distribution and dose rate based on scanning arc therapy (SPLASHLET) MATERIALS/METHODS: The algorithm optimizes (1) the clinical dose-volume constraint based on dose distribution and (2) the clinical LET-volume constraint based on LET distribution using Alternating Direction Method of Multipliers (ADMM) with Limited-memory BFGS solver by minimizing the monitor unit (MU) constraint on spot weight and (3) the effective dose-average dose rate by minimizing the accelerator's beam current sequentially. Such optimization framework enables the high dose conformal dynamic arc therapy with the capability of LET painting with voxel-based FLASH dose rate in an open-source proton planning platform (MatRad, Department of Medical Physics in Radiation Oncology, German Cancer Research Center-DKFZ). It aiming to minimize the overall cost function value combined with plan quality and voxel-based LET and dose rate constraints. Three representative cases (brain, liver and prostate cancer) were used for testing purposes. Dose-volume histogram (DVH), LET volume histogram (LVH) dose rate volume histogram (DRVH) and dose rate map were assessed compared to the original SPArc plan (SPArcoriginal). SPLASHLET plan could offer comparable plan quality compared to SPArcoriginal plan. The DRVH results indicated that SPArcoriginal could not achieve FLASH using the clinic beam current configuration, while SPLASHLET could significantly not only improve V40Gy/s in target and region of interest (ROI) but also improve the mean LET in the target and reduce the high LET in organ at risk (OAR) in comparison with SPArcoriginal (Table 1). SPLASHLET offers the first LET painting with voxel-based ultra-dose-rate and high-dose conformity treatment using proton beam therapy. Such technique has the potential to take full vantage of LET painting, FLASH and SPArc.

Inner product preconditioned trust-region methods for frequency-domain full waveform inversion

Full waveform inversion is a seismic imaging method which requires solving a large-scale minimization problem, typically through local optimization techniques. Most local optimization methods can basically be built up from two choices: the update direction and the strategy to control its length. In the context of full waveform inversion, this strategy is very often a line search. We here propose to use instead a trust-region method, in combination with non-standard inner products which act as preconditioners. More specifically, a line search and several trust-region variants of the steepest descent, the limited memory BFGS algorithm and the inexact Newton method are presented and compared. A strong emphasis is given to the inner product choice. For example, its link with preconditioning the update direction and its implication in the trust-region constraint are highlighted. A first numerical test is performed on a 2D synthetic model then a second configuration, containing two close reflectors, is studied. The latter configuration is known to be challenging because of multiple reflections. Based on these two case studies, the importance of an appropriate inner product choice is highlighted and the best trust-region method is selected and compared to the line search method. In particular we were able to demonstrate that using an appropriate inner product greatly improves the convergence of all the presented methods and that inexact Newton methods should be combined with trust-region methods to increase their convergence speed.

Adaptive CL-BFGS Algorithms for Complex-Valued Neural Networks.

Complex-valued limited-memory BFGS (CL-BFGS) algorithm is efficient for the training of complex-valued neural networks (CVNNs). As an important parameter, the memory size represents the number of saved vector pairs and would essentially affect the performance of the algorithm. However, the determination of a suitable memory size for the CL-BFGS algorithm remains challenging. To deal with this issue, an adaptive method is proposed in which the memory size is allowed to vary during the iteration process. Basically, at each iteration, with the help of multistep quasi-Newton method, an appropriate memory size is chosen from a variable set {1,2,..., M} by approximating complex Hessian matrix as close as possible. To reduce the computational complexity and ensure desired performance, the upper bound M is adjustable according to the moving average of memory sizes found in previous iterations. The proposed adaptive CL-BFGS (ACL-BFGS) algorithm can be efficiently applied for the training of CVNNs. Moreover, it is suggested to take multiple memory sizes to construct the search direction, which further improves the performance of the ACL-BFGS algorithm. Experimental results on some benchmark problems including the pattern classification, complex function approximation, and nonlinear channel equalization problems are given to illustrate the advantages of the developed algorithms over some previous ones.

Bayesian inversion with α-stable priors

We propose using Lévy α-stable distributions to construct priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices α = 1, and α = 2, respectively. Our target is to show that these priors provide a rich class of priors for modeling rough features. The main technical issue is that the α-stable probability density functions lack closed-form expressions, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. For Bayesian inversion, the currently available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate α-stable distributions that is both fast to evaluate and accurate enough from a practical viewpoint. In the numerical implementation of α-stable random field priors, we use the constructed approximation method. We show how the constructed priors can be used to solve specific Bayesian inverse problems, such as the deconvolution problem and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical α-stable priors in the one-dimensional deconvolution problem. For all numerical examples, we use maximum a posteriori estimation. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.

Multivariable optimization of pyramidal compound substrates for cooling of power-electronics in modern hybrid and electric propulsion systems

We present a method for the optimization of the thermal cooling of heat sources mounted on top of layered composites and pyramidal substrates, that are widely used in the power electronics of hybrid-electric propulsion systems. The analytical solution of the Laplace’s heat equation is approximated via Fourier expansion series and it is coupled to the Influence Coefficient Method (ICM) to provide a functional of the overall thermal stress to minimize. A multivariable optimization method is derived by coupling the equations for the heat transfer with the Sequential Least-Square Quadratic Programming (SLSQP), or the Bounded Limited-Memory BFGS (L-BFGS-B) algorithm. Code validation is performed against three-dimensional simulations and experimental data available from the literature. It is shown that an optimal component relocation and apportionment of the overall thickness of the multilayer substrate promotes a sensible reduction of the thermal stress.

Three-Dimensional Joint Inversion of the Resistivity Method and Time-Domain-Induced Polarization Based on the Cross-Gradient Constraints

The resistivity method and time-domain-induced polarization (TDIP) are two branches of electric exploration that are used to solve problems in mineral exploration, hydrogeology and engineering geology. In recent years, integrating different physical parameters for joint inversion to improve the accuracy of inversion results has been extensively examined; however, three-dimensional joint inversion of the two methods above has not been realized. To further address this issue, in this research, we used the limited-memory BFGS (L-BFGS) method to develop a three-dimensional joint inversion algorithm of the resistivity method and TDIP based on the cross-gradient constraints. In the new algorithm, the resistivity method and TDIP inversion were iteratively updated alternately to ensure that the inversion results can simultaneously meet the two conditions of obtaining minimum data misfits and finding structural similarity. The three-dimensional synthetic dataset inversion results showed that the models obtained by joint inversion are more accurate in the recovery of both the boundaries and the values of the anomalies. Especially in the background of high noise, joint inversion has higher resolution for the target body. The joint inversion algorithm was also successfully applied to a groundwater detection practice in Beijing, China, in which the practicability of the algorithm was confirmed.